12. Kant and Handedness
• "Concerning the Ultimate Foundation of the Differentiation of
Regions of Space" (1768).
Claim: Absolute space is necessary to explain the existence
of incongruent counterparts.
Immanuel Kant
(1724-1804)
[An incongruent counterpart is]...an object which is
completely like and similar to another, although it
cannot be included exactly within the same limits."
• An incongruent counterpart is a certain type of mirror image.
• Example 1:
(1)
(2)
• Maps (1) and (2) reproduce the same relations between objects.
• A relationist must say they are the same.
• An absolutist can say they are different; namely, they differ in their locations
with respect to absolute space.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
(2) O' is an incongruent counterpart of O if it cannot be made to coincide
with O by rigid motions.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
(2) O' is an incongruent counterpart of O if it cannot be made to coincide
with O by rigid motions.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
(2) O' is an incongruent counterpart of O if it cannot be made to coincide
with O by rigid motions.
Two types of mirror image
Let O be an object and let O' be its mirror image.
(1) O' is a congruent counterpart of O if it can be made to coincide with O
by rigid motions.
(2) O' is an incongruent counterpart of O if it cannot be made to coincide
with O by rigid motions.
Important Fact: Whether or not a mirror image is an incongruent
counterpart depends on the properties of the space it is located in.
• If the space is 3-dimensional, then F and its mirror image are congruent
counterparts.
• If the space is 3-dimensional, then F and its mirror image are congruent
counterparts.
• If the space is 3-dimensional, then F and its mirror image are congruent
counterparts.
• If the space is 3-dimensional, then F and its mirror image are congruent
counterparts.
• If the space is 2-dimensional and non-orientable, then F and its mirror image
are congruent counterparts.
• Example: A Möbius strip.
• If the space is 2-dimensional and non-orientable, then F and its mirror image
are congruent counterparts.
• Example: A Möbius strip.
• Obtained by identifying edge points x with x', and y with y' on a 2-dim strip.
• Result: A global "twist" that allows the mirror image of F to be rigidly
transported around the entire space back onto F.
x
•
•
y'
y
•
•
x'
• If the space is 2-dimensional and non-orientable, then F and its mirror image
are congruent counterparts.
• Example: A Möbius strip.
• Obtained by identifying edge points x with x', and y with y' on a 2-dim strip.
• Result: A global "twist" that allows the mirror image of F to be rigidly
transported around the entire space back onto F.
x
•
•
y'
y
•
•
x'
• If the space is 2-dimensional and non-orientable, then F and its mirror image
are congruent counterparts.
• Example: A Möbius strip.
• Obtained by identifying edge points x with x', and y with y' on a 2-dim strip.
• Result: A global "twist" that allows the mirror image of F to be rigidly
transported around the entire space back onto F.
x
•
•
y'
y
•
•
x'
• If the space is 2-dimensional and non-orientable, then F and its mirror image
are congruent counterparts.
• Example: A Möbius strip.
• Obtained by identifying edge points x with x', and y with y' on a 2-dim strip.
• Result: A global "twist" that allows the mirror image of F to be rigidly
transported around the entire space back onto F.
x
•
•
y'
y
•
•
x'
• If the space is 2-dimensional and non-orientable, then F and its mirror image
are congruent counterparts.
• Example: A Möbius strip.
• Obtained by identifying edge points x with x', and y with y' on a 2-dim strip.
• Result: A global "twist" that allows the mirror image of F to be rigidly
transported around the entire space back onto F.
x
•
•
y'
y
•
•
x'
• If the space is 2-dimensional and non-orientable, then F and its mirror image
are congruent counterparts.
• Example: A Möbius strip.
• Obtained by identifying edge points x with x', and y with y' on a 2-dim strip.
• Result: A global "twist" that allows the mirror image of F to be rigidly
transported around the entire space back onto F.
x
•
•
y'
y
•
•
x'
• Modified definition:
An object is an incongruent counterpart of another if
they cannot be made to occupy the same place by rigid
motions in a local (closely surrounding) region of space.
• An object is said to possess handedness (chirality) just when it and its
mirror image are incongruent counterparts.
• An object is said to lack handedness (chirality) just when it and its mirror
image are congruent counterparts.
spherical cow
• Kant's Argument for Absolute Space:
"Let it be imagined that the first created thing
were a human hand, then it must necessarily
be either a right hand or a left hand."
• But: A relationist cannot determine the handedness of an object in the
absence of other objects.
• So: Relationalism is not adequate.
• Left and right hands agree on all relational properties.
• Absolutist: They disagree on their locations with respect to absolute space.
• Letter Example Again:
• Do F and its mirror image have the same relational properties?
• Depends on how many properties one is willing to consider as relational.
THE QUICK BROWN FOX JUMPED OVER THE LAZY DOG.
THE QUICK BROWN OX JUMPED OVER THE LAZY DOG.
• F and its mirror image differ in their relational properties to the other letters
in the sentence.
• There is no way to make the mirror image of F fit into the sentence in the
same way that F does.
• Similarly, there is no way to fit a right hand
into a left-handed (Freddy Krueger) glove
(and vice versa).
• But: How can a relationist determine the handedness of an object when there
are no other reference objects to define distinguishing relational properties?
• Moreover: What if such reference objects themselves have been reflected?
• Since F and its mirror image share all relational properties in these sentences,
a relationist will not be able to distinguish them.
• Absolutist intuition: Aren't F and its mirror image distinct, independent of
their relations to other objects?
Relationist's Reflection Argument:
• Suppose: Absolute space exists.
• Then: The following two universes must be possible:
absolute space
A
absolute space
A
B
world 1 (unreflected)
Universe 1
world 2 (reflected)
Universe 2
• An absolutist must claim the reflection produces distinct worlds: They differ
on their values of absolute position.
• But: A relationist will claim that the reflection does not produce distinct
worlds: Since the relations between material objects are unaffected (and
there's no such thing as absolute space), the worlds are not distinct.
Possible Absolutist Retort:
• Would a reflected world be indiscernible from
an unreflected world?
• Replace Spock with a decaying Cobalt-60 atom:
Spock
Co60 → Ni60 + e− + νe
e−
Mirror Spock
e−
Co60
νe
Co60 decay
Co60
νe
Mirror Co60 decay
• Co60 decay (electron emitted in direction of nuclear spin) is observed more
often than Mirror Co60 decay (electron emitted in opposite direction of
nuclear spin) (Wu et al. 1957).
• Evidence that the weak force (that governs decay) violates mirror symmetry
(i.e., "parity").
absolute space
absolute space
A
A
B
world 1 (unreflected)
Universe 1
world 2 (reflected)
Universe 2
• Absolutist Claim: The reflected and unreflected worlds are not
observationally indiscernible. In world 1, the Co60 atom decay occurs more
frequently than in world 2.
• Onus is now on the relationist to explain the physical difference between
worlds 1 and 2 (recall Clarke's Dynamic Shift, with parity-violating
experiments now replacing inertial effects).
Chien-Shiung Wu
(1912-1997)
Let it be imagined that the first created thing were a Co60
decay process, then it must necessarily be either a right-handed
Co60 decay process, or a left-handed Co60 decay process... and
there's a law-like physical difference between the two!
• Can a relationist both ground the distinction between right- and left-handed
processes and explain why one is more probable than the other?
(a) Claim that the difference is intrinsic: Co60 decay processes possess an
intrinsic monadic (non-relational) property that both determines their
handedness and their weak-force-governed behavior.
(b) Claim the difference is extrinsic:
- What determines whether the first created Co60 decay process is rightor left-handed is its relation to all subsequent Co60 decay processes.
- And: It is a brute lawlike fact (in need of no further explanation) that
one of these decay processes is more probable than the other.
A Lingering Concern about Option (b):
• First: What explains Newton's First Law? How does a force-free object know
to move inertially?
- Absolutist: A local interaction between spacetime
and the object (local spacetime "feelers").
- Relationist: A nonlocal correlation between the
object and other objects (nonlocal inertial antennae).
• Similarly: What explains the parity-violating weak force? Why do Co60
atoms prefer decay modes of one chirality rather than another (given chirality
is not intrinsic)?
- Absolutist: A local interaction between spacetime
and the object.
- Relationist: A nonlocal correlation between
the object and other objects (nonlocal weakforce antennae).